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This question was motivated by the answers On noncommutative algebraic geometryOn noncommutative algebraic geometry there, he mentioned, there are some people taking category of modules as category of coherent sheaves on non-existence space. So,there might be no topological space and notions of sheaf in this settings.

My question might be related to this observation but for triangulated category. It seems that Beilinson-Bernstein take the derived category of coherent D-modules as a non-existence space, right? They used various adjoint triangle functor for these derived categories of D-modules.

So, is there a geometric space(topological space)in this framework? Is there notion of sheaf for derived category?

More general, for derived category of D-module on a scheme(not necessarily smooth), can we define topological space and sheaf for this derived category? Is there a global section functor which recover this derived category?

This question was motivated by the answers On noncommutative algebraic geometry there, he mentioned, there are some people taking category of modules as category of coherent sheaves on non-existence space. So,there might be no topological space and notions of sheaf in this settings.

My question might be related to this observation but for triangulated category. It seems that Beilinson-Bernstein take the derived category of coherent D-modules as a non-existence space, right? They used various adjoint triangle functor for these derived categories of D-modules.

So, is there a geometric space(topological space)in this framework? Is there notion of sheaf for derived category?

More general, for derived category of D-module on a scheme(not necessarily smooth), can we define topological space and sheaf for this derived category? Is there a global section functor which recover this derived category?

This question was motivated by the answers On noncommutative algebraic geometry there, he mentioned, there are some people taking category of modules as category of coherent sheaves on non-existence space. So,there might be no topological space and notions of sheaf in this settings.

My question might be related to this observation but for triangulated category. It seems that Beilinson-Bernstein take the derived category of coherent D-modules as a non-existence space, right? They used various adjoint triangle functor for these derived categories of D-modules.

So, is there a geometric space(topological space)in this framework? Is there notion of sheaf for derived category?

More general, for derived category of D-module on a scheme(not necessarily smooth), can we define topological space and sheaf for this derived category? Is there a global section functor which recover this derived category?

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Peter Lee
Peter Lee
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geometry of triangulated category and D-modules theory

This question was motivated by the answers On noncommutative algebraic geometry there, he mentioned, there are some people taking category of modules as category of coherent sheaves on non-existence space. So,there might be no topological space and notions of sheaf in this settings.

My question might be related to this observation but for triangulated category. It seems that Beilinson-Bernstein take the derived category of coherent D-modules as a non-existence space, right? They used various adjoint triangle functor for these derived categories of D-modules.

So, is there a geometric space(topological space)in this framework? Is there notion of sheaf for derived category?

More general, for derived category of D-module on a scheme(not necessarily smooth), can we define topological space and sheaf for this derived category? Is there a global section functor which recover this derived category?